Estimating the relationship between women's education and fertility in Botswana by using an instrumental variable approach to semiparametric expectile regression

We analyse the education-fertility relationship by using data on women from Botswana. A realistic quantification of such a relationship can be problematic for various reasons. First, factors such as motivation and ability are associated with fertility and education but cannot be observed and as a consequence cannot be included in the model. Here, the use of classical estimation methods will clearly result in inconsistent and biased parameter estimates. Second, there is strong heteroscedasticity in the data, which makes it very difficult to specify a suitable error distribution. Finally, covariate-response relationships can exhibit non-linear patterns. Provided that an instrumental variable is available, it is possible to employ a two-stage-type estimation approach to account for unobservable confounders. Such a technique is among the most widely used methods for isolating the effect of a predictor of interest in the presence of unobservable confounding and assures consistent estimation results. A two-stage approach can be embedded in a semiparametric expectile regression setting, hence providing possibilities for flexible additive covariate structures and modelling the whole conditional distribution of the response. Owing to its convenient estimation techniques, expectile regression may be preferable to quantile regression relying on linear programming techniques which require more numerical effort and may not accommodate very flexible model structures. We introduce a semiparametric instrumental variable expectile regression approach and study its empirical properties via an extensive simulation study. Further, corrected confidence intervals for the two-stage approach are presented. The methods are then employed to assess the education-fertility relationship.